Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. “Structure” can be understood as symmetry and a range of symmetries are expressed by hierarchy. Such symmetries directly point to invariants, that pinpoint intrinsic properties of the data and of the background empirical domain of interest. We review many aspects of hierarchy ...

In the article differential-difference (semi-discrete) lattices of hyperbolic type are investigated from integrability viewpoint. More precisely we concentrate on a method for constructing generalized symmetries. This kind integrable admit two hierarchies symmetries corresponding to discrete and continuous independent variables $n$ $x$. Symmetries direction constructed in more or less standard ...

The notion of solvable structure is generalized in order to exploit the presence an algebra symmetries for a kth-order ordinary differential equation with k > 3. In this setting, knowledg...

Abstract The YM theory has been generalized to 2YM and 3YM theories. Similarly, we generalize the BFYM “2BFYM” “3BFYM” Then, show that these higher theories can give formulations of corresponding form Finally, study gauge symmetries

Additional nonlocal symmetries of diffusion-convection equations and the Burgers equation are obtained. It is shown that these equations are connected via a generalized hodograph transformation and appropriate nonlocal symmetries arise from additional Lie symmetries of intermediate equations. Two entirely different techniques are used to search nonlocal symmetry of a given equation: the first i...